(a) It can be easily checked that \((X^\star ,\Psi ^\star ,\widehat{Y}^\star )\)  satisfying the condition is a fixed-point. Conversely, it follows from algopt1d that \(\Psi ^\star =-\nabla F(X^\star )\) , and hence \(Y^\star =\widehat{Y}^\star =X^\star \)  by algopt1a,algopt1b. Then, we obtain \((I-W)X^\star =0\)  from algopt1c. (b) It directly follows from the Karush-Kuhn-Tucker (KKT) theorem for original2 [1]}.
